In a large deep skillet or Dutch oven, melt the butter over medium heat.
Add the thinly sliced onions and salt. Cook, stirring frequently, until onions are deeply caramelized and golden brown, about 20 minutes.
Add the minced garlic and fresh thyme leaves to the onions and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and stir well to combine. Cook for 2 minutes to remove the raw flour taste.
Pour in the dry white wine and scrape the bottom of the pan to deglaze, cooking until the wine reduces by half, about 3 minutes.
Add the beef broth and whole milk to the pot and stir to combine.
Bring the mixture to a simmer, then add the uncooked pasta. Stir well to submerge the pasta in the liquid.
Cover and cook over medium-low heat, stirring occasionally, until the pasta is tender and most of the liquid is absorbed, about 12-15 minutes.
Remove the pot from heat and stir in the shredded Gruyère cheese and grated Parmesan cheese until melted and creamy.
Adjust seasoning with additional salt and pepper if needed.
Serve immediately, garnished with extra thyme if desired.